Bounding the degree of solutions to Pfaff equations
The authors investigate an old question asked by Poincaré: Given a foliation \(\mathcal F\) of degree \(d\) on an \(n\)-dimensional complex projective space and an \(\mathcal F\) -invariant projective hypersurface \(V\), is it possible to bound the degree of \(V\) by a number depending only on \(d\) and not on \(\mathcal F\) ? This question has been studied by various authors in the context of classical theory of foliations, e.g. by \textit{E. Ballico, M. Brunella, M. M. Carnicer, D. Cerveau} and \textit{A. Lins Neto}. In the present paper the authors work with Pfaff equations which are more general than foliations. Under a natural assumption that the ambient space \(X\) is a complex projective manifold with \(\text{Pic}(X)=\mathbb Z\) they give a positive answer to the above question in the case when \(V\) is a normal crossing hypersurface. At the end of the paper they discuss to which extent the normal crossing hypothesis can be weakened.
scientific article; zbMATH DE number 1563897